If you had any number of ordinary dice, what are the possible ways
of making their totals 6? What would the product of the dice be
each time?

ABC

Stage: 2 Challenge Level:

In the multiplication below, some of the digits have been
replaced by letters and others by asterisks. Where a digit has been
replaced by a letter, the same letter is used each time, and
different letters have replaced different digits. Can you
reconstruct the original multiplication?

The following splendid solution, from Prav at the North London
Collegiate School Puzzle Club, gives a clear and full explanation
of all the reasoning required to solve this problem. Very well done
Prav!

The answer is as follows:

I will now try to explain how I found my answer. We know that
when ABC is multiplied by A the answer is a three-digit number.
This means that A is an integer of value less than or equal to
three. We also know that when A is multiplied by C the resulting
answer is a number which ends in A, i.e. A * C = XA.

This means that A cannot be 1 because the only digit multiplied
by 1 to end in a number whose last digit is 1 is 1 itself and A and
C cannot be the same number. Also the value of C cannot be 1
because ABC multiplied by C would then be a 3 digit number instead
of a 4 digit number.

We know A cannot be three because there is no digit C, which
multiplied by 3, will equal X3.

Therefore our only possibility is that A is equal to two.

Because A * C = XA we can substitute A=2 into the formula. so 2
* C = X2.

The only digit which multiplied by 2 has 2 as the units digit 6,
this is because 2 * 6 = 12.

We also know that B * C = XB. We now know that C = 6 so we can
substitute this into the formula too.

B * C = XB
B * 6 = XB

The possibilities for B are 4, 6 and 8.

We know that B can not be 6 because C is 6 and both numbers can
not be the same. Also B can not be 4 because when ABC is multiplied
by B the answer is a 4-digit number but if B were 4 then we would
have 246 * 4. This results in 904, which is not a 4-digit number,
so the only possible remaining digit is 8. When substituted in the
formula it works, as shown above.

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